The Model Photosphere (Chapter The Model Photosphere (Chapter 9)9)

• Basic Assumptions• Hydrostatic Equilibrium• Temperature Distributions• Physical Conditions in Stars – the

dependence of T(), Pg(), and Pe() on effective temperature and luminosity

Basic Assumptions in Stellar AtmospheresBasic Assumptions in Stellar Atmospheres

• Local Thermodynamic Equilibrium– Ionization and excitation correctly described by

the Saha and Boltzman equations, and photon distribution is black body

• Hydrostatic Equilibrium– No dynamically significant mass loss– The photosphere is not undergoing large scale

accelerations comparable to surface gravity– No pulsations or large scale flows

• Plane Parallel Atmosphere– Only one spatial coordinate (depth)– Departure from plane parallel much larger than

photon mean free path– Fine structure is negligible (but see the Sun!)

Hydrostatic EquilibriumHydrostatic Equilibrium• Consider an element of gas

with mass dm, height dx and area dA

• The upward and downward forces on the element must balance:

PdA + gdm = (P+dP)dA• If is the density at location

x, thendm= dx dA dP/dx = g

• Since g is (nearly) constant through the atmosphere, we set

g = GM/R2

P

P+dP

x+dx

x

gdm

dP/dx = g

In Optical DepthIn Optical Depth• Since d=dx

• and dP=gdx

CLASS PROBLEM:• Recall that for a gray atmosphere,

For =0.4, Teff=104, and g=GMSun/RSun

2, compute the pressure, density, and depth at =0, ½, 2/3, 1, and 2. (The density and pressure equal zero at =0 and k =1.38 x 10-16 erg K-1)

dP/d = g/

)3

2(

4

3 44 TeffT

Estimate Teff, log g, & DepthEstimate Teff, log g, & Depth

5000 T Pe Pg 5000

1.0E-5 6896 2.56E+1 4.79E+3 2.69E-1

5.0E-4 6971 1.16E+2 7.20E+4 1.06E 0

2.0E-3 7049 2.09E+2 1.75E+5 1.81E 0

1.0E-2 7179 4.27E+2 4.74E+5 3.46E 0

4.0E-2 7379 8.93E+2 1.08E+6 6.56E 0

8.0E-2 7556 1.42E+3 1.58E+6 9.62E 0

2.0E-1 7925 3.40E+3 2.48E+6 1.79E+1

5.0E-1 8601 8.90E+3 3.58E+6 4.01E+1

8.0E-1 9114 1.74E+4 4.16E+6 6.67E+1

1.60E 0 10182 5.31E+4 4.93E+6 1.58E+2

3.0E 0 11481 1.53E+5 5.49E+6 3.89E+2

6.0E 0 13228 4.47E+5 5.94E+6 1.13E+3

Wehrse Model, Teff=10000, log g=8Wehrse Model, Teff=10000, log g=85000 T Pe Pg 5000

2.00E-03 7925 5.80E+02 8.82E+04 3.74E+00

6.00E-03 8064 9.80E+02 1.71E+05 5.48E+00

1.00E-02 8129 1.24E+03 2.33E+05 7.14E+00

2.00E-02 8208 1.70E+03 3.54E+05 9.38E+00

6.00E-02 8414 3.06E+03 6.78E+05 1.55E+01

1.00E-01 8571 4.34E+03 9.01E+05 2.06E+01

2.00E-01 8920 7.70E+03 1.28E+06 3.28E+01

5.00E-01 9600 2.03E+04 1.89E+06 7.16E+01

7.00E-01 10040 3.07E+04 2.13E+06 1.01E+02

8.00E-01 10223 3.68E+04 2.22E+06 1.18E+02

1.00E+00 10544 4.98E+04 2.37E+06 1.53E+02

1.60E+00 11377 9.77E+04 2.66E+06 2.83E+02

2.00E+00 11831 1.37E+05 2.78E+06 3.95E+02

3.00E+00 12759 2.41E+05 2.96E+06 7.15E+02

4.00E+00 13476 3.50E+05 3.07E+06 1.09E+02

6.00E+00 14278 5.01E+05 3.23E+06 1.67E+02

8.00E+00 15413 7.41E+05 3.32E+06 2.71E+02

In Integral Form -In Integral Form -

• The differential form:

• x Pg½

(where 0 is at a reference wavelength, typically 5000A)

• Then integrate:

/gd

dP

00

2

1

2

1

d

gPdPP ggg

3/2

0

log

0

2/10

3

2

000

2

1

0 loglog2

3

2

3)(

00

tde

Ptg

dt

PgP gg

g

ProcedureProcedure

• Guess at Pg()

• Guess at T()

• Do the integration, computing at each level from T and Pe

• This gives a new Pg()

• Interate until the change in Pg() is small

The T(The T() Relation) Relation

• In the Sun, we can use– Limb darkening or– The variation of with wavelength

to get the T() relation• Limb darkening can be described from:

• We have already considered limb darkening in the gray case, where

cossec),0( sec

0badeSI

baS

The Solar Limb The Solar Limb DarkeningDarkening

cossec),0( sec

0badeSI

The Solar T(The Solar T() Relation) Relation

• So one can measure I(0,) and solve for S()

• Assuming LTE (and thus setting S()=B(T)) gives us the T() relation

• The profiles of strong lines also give information about T() – different parts of a line profile are formed at different depths.

The T(The T() Relation in Other Stars) Relation in Other Stars

• Use a gray atmosphere and the Eddington approximation

• More commonly, use a scaled solar model:

• Or scale from published grid models• Comparison to T(t) relations iterated

through the equation of radiative equilibrium for flux constancy suggests scaled models are close

SunSun

TTeff

TeffT )()( *

*

Comparing T(Comparing T()’s at Teff=4000, log )’s at Teff=4000, log g=2.25g=2.25

3000

3500

4000

4500

5000

5500

6000

6500

7000

7500

0.00

1

0.00

50.

02 0.1

0.4 1 2 6 10

Optical Depth

Tem

pera

ture

Scaled HM

Bell et al.

T(T() ) vs. vs. gravitgravityy

Kurucz models at 5500K

Depart at depth, similar in shallow layers

Temperature vs. MetallicityTemperature vs. Metallicity

DON’T Scale PDON’T Scale Pgg(()!)!

3000

4000

5000

6000

7000

8000

9000

1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06

Gas Pressure

Tem

per

atu

re (

K)

Log g = 1.0

Log g = 2.0

Log g = 3.0

Log g = 4.0

Models at 5000 K

Temperature Pressure Relation Temperature Pressure Relation with Metallicitywith Metallicity

Gas Pressure vs. MetallicityGas Pressure vs. Metallicity

Electron Pressure vs. MetallicityElectron Pressure vs. Metallicity

Computing the SpectrumComputing the Spectrum

• Now can compute T, Pg, Pe, at all (Pe=NekT)

• Does the model photosphere satisfy the energy criteria (radiative equilibrium)?

• Compute the flux from

• Express I in terms of the source function S, and adopt LTE (S =B(T))

dIF cossin2 2

0